Atlas › Modular forms › Weight 2 › Level 602 › 602.2.a

Newspace 602.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(602))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
602.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	602b1 rank 0
602.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 4q^{5} - q^{7} + O(q^{8})$	+ + −	$+1$	602a1 rank 0
602.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} + q^{4} + 2q^{5} - 3q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	602c1 rank 0
602.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - 3q^{5} - 3q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
602.2.a.e	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} + q^{5} + 3q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
602.2.a.f	3	degree 3	$3q + 3q^{2} + 3q^{4} + 3q^{5} + 3q^{7} + O(q^{8})$	− − −	$+1$
602.2.a.g	3	degree 3	$3q + 3q^{2} + q^{3} + 3q^{4} + q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$
602.2.a.h	3	degree 3	$3q + 3q^{2} + 2q^{3} + 3q^{4} + 6q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	− + +	$+1$
602.2.a.i	4	degree 4	$4q - 4q^{2} + 5q^{3} + 4q^{4} - q^{5} - 5q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
602.2.a.a	$x - 5$
602.2.a.b	$x + 12$
602.2.a.c	$x - 9$
602.2.a.d	$x^{2} + 12x + 31$
602.2.a.e	$x^{3} - 61x - 32$
602.2.a.f	$x^{3} - 9x^{2} - 5x + 61$
602.2.a.g	$x^{3} - x^{2} - 180x + 208$
602.2.a.h	$x^{3} - 20x^{2} + 127x - 250$
602.2.a.i	$x^{4} - 2x^{3} - 190x^{2} - 166x + 5869$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(602))$	21
Old forms	64
Cusp forms $S_{2}(\Gamma_0(602))$	85
Eisenstein series $E_{2}(\Gamma_0(602))$	7
Modular forms $M_{2}(\Gamma_0(602))$	92

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(602)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(86))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(301))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.247 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 2, 7, 14, 43, 86, 301, 602]: [0, 1, 1, 4, 4, 13, 30, 92] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 3 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(602, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 602:2 1000

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